Question: Simplify the following expression: $y = \dfrac{7x^2+24x- 16}{7x - 4}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-16)} &=& -112 \\ {a} + {b} &=& &=& {24} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-112$ and add them together. Remember, since $-112$ is negative, one of the factors must be negative. The factors that add up to ${24}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${28}$ $ \begin{eqnarray} {ab} &=& ({-4})({28}) &=& -112 \\ {a} + {b} &=& {-4} + {28} &=& 24 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({7}x^2 {-4}x) + ({28}x {-16}) $ Factor out the common factors: $ x(7x - 4) + 4(7x - 4)$ Now factor out $(7x - 4)$ $ (7x - 4)(x + 4)$ The original expression can therefore be written: $ \dfrac{(7x - 4)(x + 4)}{7x - 4}$ We are dividing by $7x - 4$ , so $7x - 4 \neq 0$ Therefore, $x \neq \frac{4}{7}$ This leaves us with $x + 4; x \neq \frac{4}{7}$.